Statistical Decision Theory by Nicholas T. Longford

Author:Nicholas T. Longford
Language: eng
Format: epub
Publisher: Springer Berlin Heidelberg, Berlin, Heidelberg

Inferences related exclusively to or are based on the corresponding marginal posterior distributions. By extracting from the density of , we obtain the marginal posterior distribution of :

where is the constant for which the right-hand side is a density. Thus, the posterior distribution of is inverse gamma, with shape and scale .

The marginal posterior of is derived by integrating the joint posterior density over ;

(4.9)

obtained by matching the density , as a function of , with fixed, to the density of an inverse gamma distribution. Denote by the density of the distribution with degrees of freedom,

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